Advertisements
Advertisements
Question
\[e^x \log \sqrt{x} \tan x\]
Advertisements
Solution
\[\text{ Let } u = e^x ; v = \log \sqrt{x}; w = \tan x\]
\[\text{ Then } , u' = e^x ; v' = \frac{1}{\sqrt{x}} \times \frac{1}{2\sqrt{x}} = \frac{1}{2x}; w' = \sec^2 x\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uvw \right) = u'vw + uv'w + uvw'\]
\[ = e^x \log \sqrt{x}\tan x + e^x \times \frac{1}{2x}\tan x + e^x \log \sqrt{x} \sec^2 x\]
\[ = e^x \left( \log x^\frac{1}{2} . \tan x + \frac{\tan x}{2x} + \log x^\frac{1}{2} . \sec^2 x \right)\]
\[ = e^x \left( \frac{1}{2} \log x . \tan x + \frac{\tan x}{2x} + \frac{1}{2} \log x . \sec^2 x \right)\]
\[ = \frac{e^x}{2}\left( \log x . \tan x + \frac{\tan x}{x} + \log x . \sec^2 x \right)\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x5 (3 – 6x–9).
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(px+ q) (r/s + s)`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
sinn x
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
x4 (5 sin x – 3 cos x)
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
k xn
(x2 + 1) (x − 5)
\[\sqrt{2 x^2 + 1}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate each of the following from first principle:
e−x
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x3 sin x
xn tan x
xn loga x
sin x cos x
\[\frac{2^x \cot x}{\sqrt{x}}\]
(x sin x + cos x) (x cos x − sin x)
(1 − 2 tan x) (5 + 4 sin x)
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3x2 + 2)2
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{1}{a x^2 + bx + c}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
